5.3 The sine, cosine and tangent functions: graphs and periodicity — Practise with help

Write down the period, the amplitude and the range before you draw anything. The shape of the wave tells you most of what a question needs.

Warm-up: remember

Write down the value of $\sin 0$ and $\cos 0$ .
Write down the value of $\sin\dfrac{\pi}{2}$ and $\cos\dfrac{\pi}{2}$ .
Write down the value of $\tan\dfrac{\pi}{4}$ .

Answers

$\sin 0 = 0$ and $\cos 0 = 1$ .
$\sin\dfrac{\pi}{2} = 1$ and $\cos\dfrac{\pi}{2} = 0$ .
$\tan\dfrac{\pi}{4} = 1$ .

If question 2 was hard, revisit the unit circle before you continue.

Practise with help

1. State the amplitude, the period and the range of $y = 4\sin x$ .

Hint

Compare the equation with $y = A\sin(Bx + C) + D$ . Here $B = 1$ and $D = 0$ .

Next step

The amplitude is the number in front of the sine. The period is $\dfrac{2\pi}{B}$ with $B = 1$ .

Full working

Amplitude $= 4$ . Period $= \dfrac{2\pi}{1} = 2\pi$ . The range is $-4 \le y \le 4$ , because the middle line is the $x$ -axis.

2. State the amplitude, the period and the range of $y = 2\cos(4x) + 3$ .

Hint

Identify $A$ , $B$ and $D$ first. The middle line is $y = D$ .

Next step

Here $A = 2$ , $B = 4$ and $D = 3$ . The period is $\dfrac{2\pi}{4}$ , and the range runs from $D - |A|$ to $D + |A|$ .

Full working

Amplitude $= 2$ . Period $= \dfrac{2\pi}{4} = \dfrac{\pi}{2}$ . The middle line is $y = 3$ , so the range is $1 \le y \le 5$ .

3. Write down the period and the phase shift of $y = \sin\left(2x - \dfrac{\pi}{3}\right)$ .

Hint

Match the bracket with $Bx + C$ . The phase shift is $-\dfrac{C}{B}$ .

Next step

Here $B = 2$ and $C = -\dfrac{\pi}{3}$ . Work out the period $\dfrac{2\pi}{2}$ , then the phase shift $-\dfrac{-{\pi}/{3}}{2}$ .

Full working

Period $= \dfrac{2\pi}{2} = \pi$ . Phase shift $= -\dfrac{C}{B} = \dfrac{\pi}{6}$ to the right.

4. Sketch one full period of $y = \tan x$ . Mark the asymptotes and the points where the curve crosses the $x$ -axis.

Hint

The period of tangent is $\pi$ . The asymptotes sit where $\cos x = 0$ .

Next step

On the interval $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ , mark the asymptotes at the two ends and the crossing at $x = 0$ .

Full working

The curve rises from below through $(0, 0)$ and climbs without limit towards the asymptote at $x = \dfrac{\pi}{2}$ . It comes up from $-\infty$ just after $x = -\dfrac{\pi}{2}$ . The same shape repeats every $\pi$ . The curve crosses the $x$ -axis at $x = 0$ and at every multiple of $\pi$ .

5. The height of a point on a wheel above its centre is $h = 1.5\sin(0.8t)$ metres, where $t$ is in seconds. Find the greatest height and the period of the motion.

Hint

The greatest height is the amplitude. The period comes from $B = 0.8$ .

Next step

The amplitude is $1.5$ , so the highest point is $1.5$ m above the centre. The period is $\dfrac{2\pi}{0.8}$ .

Full working

Greatest height $= 1.5$ m. Period $= \dfrac{2\pi}{0.8} = 2.5\pi \approx 7.85$ s. The point returns to the same height every $2.5\pi$ seconds.

6. The depth of water at a harbour is modelled by $d = 3 + 2\sin\left(\dfrac{\pi t}{6}\right)$ metres, where $t$ is in hours. State the greatest and least depth, and the time for one full cycle.

Hint

The middle line is $D = 3$ and the amplitude is $A = 2$ .

Next step

The greatest depth is $3 + 2$ and the least is $3 - 2$ . The period is $\dfrac{2\pi}{\pi/6}$ .

Full working

Greatest depth $= 5$ m. Least depth $= 1$ m. Period $= \dfrac{2\pi}{\pi/6} = 12$ hours. The sea returns to the same depth every $12$ hours.